ISSN:
1420-8911
Keywords:
Key words and phrases: Boolean algebra, $ \alpha $-cut-completion, $ \alpha $-injective, $ \alpha $-cloz space, quasi- $ F_\alpha $ space.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. Let A be a Boolean algebra, and $ \alpha $ an infinite cardinal number or the symbol $ \infty $ . An $ \alpha $ -cut in A is an ordered pair (F,H) of subsets of A, each of power $ \le \alpha $ , with $ F \le H $ elementwise, with 0 as the meet of differences $ h - f (h \in H, f \in F) $ . A is called $ \alpha $ -cut-complete if for each $ \alpha $ -cut (F,H) there is $ a \in A $ with $ F \le a \le H $ elementwise. We describe the simply-constructed $ \alpha $ -cut-completion $ A^\alpha $ , show that $ \alpha $ -cut-completeness solves a natural $ \alpha $ -injectivity problem, determine when $ A^\alpha $ is the $ \alpha $ -completion, or the completion, and interpret most of that topologically in Stone spaces. Oddly, these considerations seem novel in Boolean algebras, while for lattice-ordered groups and vector lattices, and dually for topological spaces, the analogous theory, especially for $ \alpha = \omega_1 $ , has received considerable study.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000120050067
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